In Taiwan, what is the penalty for overstaying a tourist visa with legitimate reasons (for example, a flight was cancelled or delayed, or a transportation closure made someone miss getting to the airport in time)? What fines or barriers to future travel would someone face?

As mentioned by Michael Pryor, there is a fine. Any natural force is still your responsibility, including typhoons. From my understanding, the only way you can overstay your visa is by government mandate (quarantine, jail, etc).

That being said, I have overstayed a visa by 2 days and didn't have to pay anything because there was a typhoon. Really, it is up to the discretion of the immigration officer you meet. Also note, that you can be jailed for up to 2 MONTHS if you overstay. I have never heard of that being used except in the case of a suspected drug smuggler though . Your best bet is to check the weather forecast. If you absolutely can't get out on the last day of your visa, just accept the fact that you will have to pay some cash.